On this note, we here for conceptual (and calculational [1]}, [2]}) clarity interpreted the CVE exclusively from the rotation \({\Omega }\)  in the preferred frame with torsion. For consistency of the background, the CTE and CVE should be possible to understand solely in terms of the tetrad/metric, where the both CVE source \({v}_{n}\)  and (torsionful) \({v}_s\)  tetrad directly contribute. In case of torsion, metric variations are superseded by those corresponding to the tetrad and connection. Moreover, Galilean invariance immediately suggests that \({v}_s-{v}_n\)  should enter as the non-trivial component. Moreover, the non-relativistic extension of CTE could explain the \(T=0\)  anomalous angular momentum terms in the chiral \(p+ip\)  system [3]}. Other non-relativistic systems where torsion is directly relevant are Weyl semimetals with dislocations: in the continuum approximation, the dislocation density arises from the torsion of globally non-trivial tetrad variation \(\delta e^a_{\mu } = \partial _{\mu }u^a\) , where \(u^a\)  is now the elastic displacement. In the case where \(u^a\)  is globally continuous, the non-trivial field strength needed for tetrads (or strain pseudo gauge fields) vanishes. In fact, \(T^a_{\mu \nu }\)  is equivalent to “spatial vorticity" of the \(u^a\)  from dislocations and the findings of this paper can be applied with minimum alterations to axially twisted semimetals. What is missing is the spatial analogy to the static, equilibrium CVE and CTE is expected to play its role. In addition to twisting, the corresponding bias for dislocations could be the real geometry/gravitational field, in the absence of rotation [4]}. Nevertheless, see [5]} for some recent considerations of the anomalous transport and CME without dislocations (and therefore no effective torsion).
